1738:(two less than a power of two). The question is almost completely resolved: there are manifolds with nonzero Kervaire invariant in dimension 2, 6, 14, 30, 62, and none in all other dimensions other than possibly 126. However, Zhouli Xu (in collaboration with Weinan Lin and Guozhen Wang) announced on May 30, 2024 that there exists a manifold with nonzero Kervaire invariant in dimension 126.
494:
1901:
in dimension 64, and the octo-octonionic projective plane in dimension 128), specifically that there is a construction that takes these projective planes and produces a manifold with nonzero
Kervaire invariant in two dimensions lower.
1857:
that there is such a manifold in dimension 126, and that the higher-dimensional manifolds with nonzero
Kervaire invariant are related to well-known exotic manifolds two dimension higher, in dimensions 16, 32, 64, and 128, namely the
1463:. The quotients are the difficult parts of the groups. The map between these quotient groups is either an isomorphism or is injective and has an image of index 2. It is the latter if and only if there is an
286:
In any given dimension, there are only two possibilities: either all manifolds have Arf–Kervaire invariant equal to 0, or half have Arf–Kervaire invariant 0 and the other half have Arf–Kervaire invariant 1.
1533:
942:
1245:
2089:
2054:
395:
819:
2149:
1113:(in dimension greater than 4), with one step in the computation depending on the Kervaire invariant problem. Specifically, they show that the set of exotic spheres of dimension
867:
The
Kervaire invariant is a generalization of the Arf invariant of a framed surface (that is, a 2-dimensional manifold with stably trivialized tangent bundle) which was used by
202:
1891:
1622:
302:(in collaboration with Weinan Lin and Guozhen Wang), announced during a seminar at Princeton University that the final case of dimension 126 has been settled. Xu stated that
1350:
1146:
745:
1467:-dimensional framed manifold of nonzero Kervaire invariant, and thus the classification of exotic spheres depends up to a factor of 2 on the Kervaire invariant problem.
1848:
332:
2332:
1663:
1402:
1284:
707:
653:
617:
563:
281:
129:
1822:
1784:
1736:
1079:
1046:
975:
857:
1457:
241:
56:
1020:
994:
672:
582:
403:
2850:
1318:
1560:
1103:, the first example of such a manifold, by showing that his invariant does not vanish on this PL manifold, but vanishes on all smooth manifolds of dimension 10.
156:
1422:
2590:
2251:
2216:
Anderson, Donald W.; Brown, Edgar H. Jr.; Peterson, Franklin P. (January 1966). "SU-corbodism, KO-characteristic
Numbers, and the Kervaire Invariant".
2017:
invariant is a closely related invariant of framed surgery of a 2, 6 or 14-dimensional framed manifold, that gives isomorphisms from the 2nd and 6th
66:
converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphere, and 1 otherwise. This invariant was named after
1998:
The coefficient groups Ω(point) can detect non-vanishing
Kervaire invariants: more precisely if the Kervaire invariant for manifolds of dimension
2697:
2340:
343:
337:
2793:
2823:
Hypersphere
Exotica: Kervaire Invariant Problem Has a Solution! A 45-year-old problem on higher-dimensional spheres is solved–probably
1482:
2424:
1898:
877:
2018:
1357:
2324:
2290:
1742:
1169:
2866:
2809:
515:
on the middle dimensional homology of an (unframed) even-dimensional manifold; the framing yields the quadratic refinement.
2669:
2651:
2368:
1960:
showed that the
Kervaire invariant is nonzero for some manifold of dimension 62. An alternative proof was given later by
1982:≥ 8. They constructed a cohomology theory Ω with the following properties from which their result follows immediately:
2664:
2646:
2363:
2469:
1894:
2803:
2375:
298:, this can happen in dimensions 2, 6, 14, 30, 62, and possibly 126, and in no other dimensions. On May 30, 2024,
2059:
2024:
1850:). Together with explicit constructions for lower dimensions (through 62), this leaves open only dimension 126.
365:
2659:
754:
2681:
2871:
1859:
1291:
527:
295:
21:
2098:
2260:
1750:
2641:
2612:
2388:
2295:
161:
2822:
1864:
344:
https://www.math.princeton.edu/events/computing-differentials-adams-spectral-sequence-2024-05-30t170000
2729:
2567:
2160:
1565:
508:
132:
2265:
2827:
2358:
619:
determined by the framing, or by the triviality/non-triviality of the normal bundles of embeddings
512:
504:
82:
1323:
2752:
2719:
2629:
2557:
2521:
2492:
2449:
2397:
2379:
2312:
2233:
1787:
1124:
712:
1827:
305:
2693:
2336:
2170:
1627:
1371:
1253:
1092:
676:
622:
586:
532:
250:
205:
98:
1801:
1756:
1708:
1051:
1025:
947:
824:
489:{\displaystyle q\colon H_{2m+1}(M;\mathbb {Z} /2\mathbb {Z} )\to \mathbb {Z} /2\mathbb {Z} ,}
294:
is the problem of determining in which dimensions the
Kervaire invariant can be nonzero. For
2839:
2762:
2685:
2621:
2530:
2484:
2433:
2407:
2304:
2270:
2225:
1536:
1460:
1427:
1100:
211:
26:
2774:
2707:
2544:
2504:
2445:
2350:
2282:
998:
2770:
2703:
2607:
2603:
2540:
2500:
2461:
2441:
2419:
2383:
2346:
2278:
1791:
1424:
have easily understood cyclic factors, which are trivial or order two except in dimension
1156:
979:
657:
567:
523:
67:
2194:
1297:
2733:
2571:
1542:
138:
2582:
1854:
1674:
1666:
1407:
1365:
872:
868:
748:
359:
86:
63:
2249:(1984). "Relations amongst Toda brackets and the Kervaire invariant in dimension 62".
2860:
2535:
2516:
2512:
2453:
2246:
1670:
1110:
355:
78:
2203:
334:
survives so that there exists a manifold of
Kervaire invariant 1 in dimension 126.
1920:
proved that the
Kervaire invariant can be nonzero for manifolds of dimension 6, 14
2578:
2465:
2411:
1149:
1096:
2274:
1940:
proved that the Kervaire invariant can be nonzero for manifolds of dimension 30
2788:
2689:
2293:(1969). "The Kervaire invariant of framed manifolds and its generalization".
2806:, April 23, 2009, blog post by John Baez and discussion, The n-Category Café
2766:
2740:
1914:
proved that the Kervaire invariant is zero for manifolds of dimension 10, 18
299:
71:
2554:
Kervaire Invariant One (after M. A. Hill, M. J. Hopkins, and D. C. Ravenel)
2843:
2056:, and a homomorphism from the 14th stable homotopy group of spheres onto
1693:-dimensional framed manifolds of nonzero Kervaire invariant is called the
2798:
93:
59:
2422:(1960). "A manifold which does not admit any differentiable structure".
2633:
2496:
2437:
2316:
2237:
1926:
proved that the Kervaire invariant is zero for manifolds of dimension 8
1946:
proved that the Kervaire invariant is zero for manifolds of dimension
2386:(2016). "On the nonexistence of elements of Kervaire invariant one".
2625:
2488:
2308:
2229:
2198:
2789:
Slides and video of lecture by Hopkins at Edinburgh, 21 April, 2009
1677:
1), and thus the standard embedded torus has Kervaire invariant 0.
2757:
2724:
2562:
2402:
1476:
2743:(2016), "The Strong Kervaire invariant problem in dimension 62",
1985:
The coefficient groups Ω(point) have period 2 = 256 in
2335:, vol. 65, New York-Heidelberg: Springer, pp. ix+132,
131:, and thus analogous to the other invariants from L-theory: the
1528:{\displaystyle {\begin{pmatrix}0&1\\-1&0\end{pmatrix}}}
1117:– specifically the monoid of smooth structures on the standard
1991:
The coefficient groups Ω(point) have a "gap": they vanish for
1893:(dimension 16, octonionic projective plane) and the analogous
937:{\displaystyle \pi _{n+2}(S^{n})=\mathbb {Z} /2\mathbb {Z} }
2199:
Kervaire invariant: Why dimension 126 especially difficult?
2851:
Mathematicians solve 45-year-old Kervaire invariant puzzle
2224:(1). Mathematics Department, Princeton University: 54–67.
1459:, in which case they are large, with order related to the
1897:(the bi-octonionic projective plane in dimension 32, the
1749:), who reduced the problem from differential topology to
1240:{\displaystyle \Theta _{n}/bP_{n+1}\to \pi _{n}^{S}/J,\,}
1048:), which is the cobordism group of surfaces embedded in
338:"Computing differentials in the Adams spectral sequence"
2834:
Ball, Philip (2009). "Hidden riddle of shapes solved".
2002:
is nonzero then it has a nonzero image in Ω(point)
158:-dimensional invariant (either symmetric or quadratic,
1491:
522:
can be defined by algebraic topology using functional
2101:
2062:
2027:
1923:
1867:
1830:
1804:
1759:
1711:
1630:
1568:
1545:
1485:
1430:
1410:
1374:
1326:
1300:
1256:
1172:
1127:
1054:
1028:
1001:
982:
950:
880:
827:
757:
715:
679:
660:
625:
589:
570:
535:
406:
368:
308:
253:
214:
164:
141:
101:
29:
2799:
Harvard-MIT Summer Seminar on the Kervaire Invariant
1957:
1798:), who showed that there were no such manifolds for
362:
determined by the framing on the middle-dimensional
2517:"Some differentials in the Adams spectral sequence"
526:, and geometrically via the self-intersections of
2143:
2083:
2048:
1967:
1885:
1842:
1816:
1795:
1778:
1730:
1657:
1616:
1554:
1527:
1451:
1416:
1396:
1344:
1312:
1278:
1239:
1140:
1073:
1040:
1014:
988:
969:
936:
851:
813:
739:
701:
666:
647:
611:
576:
557:
488:
389:
326:
275:
235:
196:
150:
123:
50:
2678:Stable homotopy around the Arf-Kervaire invariant
1753:and showed that the only possible dimensions are
1539:), and the skew-quadratic refinement is given by
2333:Ergebnisse der Mathematik und ihrer Grenzgebiete
1970:showed that the Kervaire invariant is zero for
1937:
89:. It can be thought of as the simply-connected
2716:The Arf-Kervaire Invariant of framed manifolds
1917:
1106:
2583:"Differential topology forty-six years later"
2197:by André Henriques Jul 1, 2012 at 19:26, on "
1163:. They compute this latter in terms of a map
8:
2591:Notices of the American Mathematical Society
1673:0 (most of its elements have norm 0; it has
62:that measures whether the manifold could be
1368:, which is also a cyclic group. The groups
2252:Journal of the London Mathematical Society
2084:{\displaystyle \mathbb {Z} /2\mathbb {Z} }
2049:{\displaystyle \mathbb {Z} /2\mathbb {Z} }
390:{\displaystyle \mathbb {Z} /2\mathbb {Z} }
2756:
2723:
2561:
2534:
2401:
2264:
2131:
2127:
2110:
2106:
2100:
2095:= 2, 6, 14 there is an exotic framing on
2077:
2076:
2068:
2064:
2063:
2061:
2042:
2041:
2033:
2029:
2028:
2026:
1877:
1868:
1866:
1829:
1803:
1764:
1758:
1716:
1710:
1629:
1567:
1544:
1486:
1484:
1429:
1409:
1382:
1373:
1336:
1331:
1325:
1299:
1264:
1255:
1236:
1225:
1219:
1214:
1195:
1183:
1177:
1171:
1132:
1126:
1059:
1053:
1027:
1006:
1000:
981:
955:
949:
930:
929:
921:
917:
916:
904:
885:
879:
826:
790:
762:
756:
714:
684:
678:
659:
630:
624:
594:
588:
569:
540:
534:
479:
478:
470:
466:
465:
455:
454:
446:
442:
441:
417:
405:
383:
382:
374:
370:
369:
367:
318:
313:
307:
258:
252:
213:
185:
169:
163:
140:
106:
100:
77:The Kervaire invariant is defined as the
28:
1911:
1786:, and those of Michael A. Hill,
1624:: the basis curves don't self-link; and
1084:
814:{\displaystyle S^{4m+2+k}\to S^{2m+1+k}}
2804:'Kervaire Invariant One Problem' Solved
2610:(1971), "On the Kervaire obstruction",
2187:
1943:
1746:
2825:, by Davide Castelvecchi, August 2009
2794:Arf-Kervaire home page of Doug Ravenel
2245:Barratt, Michael G.; Jones, J. D. S.;
1479:, the skew-symmetric form is given by
2329:Surgery on simply-connected manifolds
2177: + 1)-dimensional invariant
2144:{\displaystyle S^{n/2}\times S^{n/2}}
1924:Anderson, Brown & Peterson (1966)
1701:is 2 mod 4, and indeed one must have
1121:-sphere – is isomorphic to the group
7:
2684:, vol. 273, Birkhäuser Verlag,
1958:Barratt, Jones & Mahowald (1984)
1950:not of the form 2 −
1685:The question of in which dimensions
2151:with Kervaire–Milnor invariant 1.
1974:-dimensional framed manifolds for
1968:Hill, Hopkins & Ravenel (2016)
1961:
1174:
1129:
197:{\displaystyle L^{4k}\cong L_{4k}}
14:
2425:Commentarii Mathematici Helvetici
1899:quateroctonionic projective plane
1886:{\displaystyle \mathbf {O} P^{2}}
1091: = 10 to construct the
499:and is thus sometimes called the
2019:stable homotopy group of spheres
1869:
1665:: a (1,1) self-links, as in the
1358:stable homotopy group of spheres
1081:with trivialized normal bundle.
503:. The quadratic form (properly,
2470:"Groups of homotopy spheres: I"
1617:{\displaystyle Q(1,0)=Q(0,1)=0}
2853:, Erica Klarreich, 20 Jul 2009
1741:The main results are those of
1646:
1634:
1605:
1593:
1584:
1572:
1535:(with respect to the standard
1207:
983:
910:
897:
783:
661:
571:
462:
459:
432:
354:The Kervaire invariant is the
230:
215:
45:
30:
1:
1938:Mahowald & Tangora (1967)
2536:10.1016/0040-9383(67)90023-7
1918:Kervaire & Milnor (1963)
1562:with respect to this basis:
1345:{\displaystyle \pi _{n}^{S}}
1107:Kervaire & Milnor (1963)
397:-coefficient homology group
2665:Encyclopedia of Mathematics
2660:"Kervaire-Milnor invariant"
2647:Encyclopedia of Mathematics
2412:10.4007/annals.2016.184.1.1
2364:Encyclopedia of Mathematics
2357:Chernavskii, A.V. (2001) ,
1895:Rosenfeld projective planes
1697:. This is only possible if
1141:{\displaystyle \Theta _{n}}
740:{\displaystyle m\neq 0,1,3}
336:Xu, Zhouli (May 30, 2024).
2888:
2714:Snaith, Victor P. (2010),
2676:Snaith, Victor P. (2009),
2515:; Tangora, Martin (1967).
1695:Kervaire invariant problem
1681:Kervaire invariant problem
1475:For the standard embedded
1286:is the cyclic subgroup of
292:Kervaire invariant problem
85:on the middle dimensional
2690:10.1007/978-3-7643-9904-7
2009:Kervaire–Milnor invariant
1843:{\displaystyle n\geq 254}
327:{\displaystyle h_{6}^{2}}
2658:Shtan'ko, M.A. (2001) ,
2640:Shtan'ko, M.A. (2001) ,
2552:Miller, Haynes (2012) ,
2275:10.1112/jlms/s2-30.3.533
1658:{\displaystyle Q(1,1)=1}
1397:{\displaystyle bP_{n+1}}
1279:{\displaystyle bP_{n+1}}
1101:differentiable structure
702:{\displaystyle M^{4m+2}}
648:{\displaystyle S^{2m+1}}
612:{\displaystyle M^{4m+2}}
558:{\displaystyle S^{2m+1}}
296:differentiable manifolds
276:{\displaystyle L^{4k+1}}
124:{\displaystyle L_{4k+2}}
2767:10.2140/gt.2016.20.1611
2745:Geometry & Topology
2682:Progress in Mathematics
1860:Cayley projective plane
1817:{\displaystyle k\geq 8}
1779:{\displaystyle 2^{k}-2}
1731:{\displaystyle 2^{k}-2}
1292:parallelizable manifold
1087:used his invariant for
1074:{\displaystyle S^{n+2}}
1041:{\displaystyle n\geq 2}
970:{\displaystyle S^{n+2}}
871:in 1950 to compute the
852:{\displaystyle m=0,1,3}
2556:, Seminaire Bourbaki,
2167:-dimensional invariant
2145:
2085:
2050:
1887:
1853:It was conjectured by
1844:
1818:
1780:
1751:stable homotopy theory
1732:
1659:
1618:
1556:
1529:
1453:
1452:{\displaystyle n=4k+3}
1418:
1398:
1346:
1314:
1290:-spheres that bound a
1280:
1241:
1142:
1109:computes the group of
1075:
1042:
1016:
990:
971:
938:
853:
815:
741:
703:
668:
649:
613:
578:
559:
501:Arf–Kervaire invariant
490:
391:
328:
277:
237:
236:{\displaystyle (4k+1)}
198:
152:
125:
52:
51:{\displaystyle (4k+2)}
2867:Differential topology
2844:10.1038/news.2009.427
2812:at the manifold atlas
2613:Annals of Mathematics
2477:Annals of Mathematics
2389:Annals of Mathematics
2296:Annals of Mathematics
2218:Annals of Mathematics
2146:
2086:
2051:
1888:
1845:
1819:
1781:
1733:
1669:. This form thus has
1660:
1619:
1557:
1530:
1454:
1419:
1399:
1347:
1315:
1281:
1242:
1143:
1076:
1043:
1017:
1015:{\displaystyle S^{n}}
991:
972:
939:
854:
816:
742:
704:
669:
650:
614:
579:
560:
491:
392:
329:
278:
238:
199:
153:
126:
70:who built on work of
53:
20:is an invariant of a
2817:Popular news stories
2642:"Kervaire invariant"
2099:
2060:
2025:
1865:
1828:
1802:
1757:
1709:
1628:
1566:
1543:
1483:
1428:
1408:
1372:
1364:is the image of the
1324:
1298:
1254:
1170:
1155:classes of oriented
1125:
1052:
1026:
999:
989:{\displaystyle \to }
980:
948:
878:
825:
755:
713:
677:
667:{\displaystyle \to }
658:
623:
587:
577:{\displaystyle \to }
568:
533:
509:quadratic refinement
404:
366:
306:
251:
212:
162:
139:
99:
27:
16:In mathematics, the
2828:Scientific American
2734:2010arXiv1001.4751S
2608:Sullivan, Dennis P.
2572:2011arXiv1104.4523M
2462:Kervaire, Michel A.
2420:Kervaire, Michel A.
2384:Ravenel, Douglas C.
2380:Hopkins, Michael J.
1743:William Browder
1341:
1313:{\displaystyle n+1}
1224:
1095:, a 10-dimensional
518:The quadratic form
505:skew-quadratic form
323:
83:skew-quadratic form
2438:10.1007/bf02565940
2141:
2081:
2046:
1978:= 2− 2 with
1883:
1840:
1814:
1792:Douglas C. Ravenel
1788:Michael J. Hopkins
1776:
1728:
1655:
1614:
1555:{\displaystyle xy}
1552:
1525:
1519:
1449:
1414:
1394:
1342:
1327:
1310:
1276:
1237:
1210:
1138:
1071:
1038:
1012:
986:
967:
934:
849:
811:
737:
699:
664:
645:
609:
574:
555:
486:
387:
324:
309:
273:
233:
194:
151:{\displaystyle 4k}
148:
121:
48:
18:Kervaire invariant
2699:978-3-7643-9903-0
2513:Mahowald, Mark E.
2342:978-0-387-05629-6
2247:Mahowald, Mark E.
2220:. Second Series.
2171:De Rham invariant
1461:Bernoulli numbers
1417:{\displaystyle J}
1093:Kervaire manifold
206:De Rham invariant
2879:
2847:
2777:
2760:
2751:(3): 1611–1624,
2736:
2727:
2710:
2672:
2654:
2636:
2604:Rourke, Colin P.
2599:
2587:
2574:
2565:
2548:
2538:
2508:
2474:
2457:
2415:
2405:
2376:Hill, Michael A.
2371:
2353:
2325:Browder, William
2320:
2291:Browder, William
2286:
2268:
2241:
2207:
2192:
2150:
2148:
2147:
2142:
2140:
2139:
2135:
2119:
2118:
2114:
2090:
2088:
2087:
2082:
2080:
2072:
2067:
2055:
2053:
2052:
2047:
2045:
2037:
2032:
1995:= -1, -2, and -3
1892:
1890:
1889:
1884:
1882:
1881:
1872:
1849:
1847:
1846:
1841:
1823:
1821:
1820:
1815:
1785:
1783:
1782:
1777:
1769:
1768:
1737:
1735:
1734:
1729:
1721:
1720:
1664:
1662:
1661:
1656:
1623:
1621:
1620:
1615:
1561:
1559:
1558:
1553:
1537:symplectic basis
1534:
1532:
1531:
1526:
1524:
1523:
1458:
1456:
1455:
1450:
1423:
1421:
1420:
1415:
1403:
1401:
1400:
1395:
1393:
1392:
1351:
1349:
1348:
1343:
1340:
1335:
1319:
1317:
1316:
1311:
1285:
1283:
1282:
1277:
1275:
1274:
1246:
1244:
1243:
1238:
1229:
1223:
1218:
1206:
1205:
1187:
1182:
1181:
1147:
1145:
1144:
1139:
1137:
1136:
1080:
1078:
1077:
1072:
1070:
1069:
1047:
1045:
1044:
1039:
1021:
1019:
1018:
1013:
1011:
1010:
995:
993:
992:
987:
976:
974:
973:
968:
966:
965:
943:
941:
940:
935:
933:
925:
920:
909:
908:
896:
895:
858:
856:
855:
850:
820:
818:
817:
812:
810:
809:
782:
781:
747:) and the mod 2
746:
744:
743:
738:
708:
706:
705:
700:
698:
697:
673:
671:
670:
665:
654:
652:
651:
646:
644:
643:
618:
616:
615:
610:
608:
607:
583:
581:
580:
575:
564:
562:
561:
556:
554:
553:
524:Steenrod squares
513:ε-symmetric form
495:
493:
492:
487:
482:
474:
469:
458:
450:
445:
431:
430:
396:
394:
393:
388:
386:
378:
373:
341:
333:
331:
330:
325:
322:
317:
282:
280:
279:
274:
272:
271:
242:
240:
239:
234:
203:
201:
200:
195:
193:
192:
177:
176:
157:
155:
154:
149:
130:
128:
127:
122:
120:
119:
57:
55:
54:
49:
2887:
2886:
2882:
2881:
2880:
2878:
2877:
2876:
2857:
2856:
2833:
2819:
2785:
2780:
2739:
2713:
2700:
2675:
2657:
2639:
2626:10.2307/1970764
2602:
2585:
2579:Milnor, John W.
2577:
2551:
2511:
2489:10.2307/1970128
2472:
2466:Milnor, John W.
2460:
2418:
2374:
2359:"Arf invariant"
2356:
2343:
2323:
2309:10.2307/1970686
2289:
2266:10.1.1.212.1163
2244:
2230:10.2307/1970470
2215:
2211:
2210:
2193:
2189:
2184:
2157:
2123:
2102:
2097:
2096:
2058:
2057:
2023:
2022:
2015:Kervaire–Milnor
2011:
1912:Kervaire (1960)
1908:
1873:
1863:
1862:
1826:
1825:
1800:
1799:
1760:
1755:
1754:
1712:
1707:
1706:
1705:is of the form
1683:
1626:
1625:
1564:
1563:
1541:
1540:
1518:
1517:
1512:
1503:
1502:
1497:
1487:
1481:
1480:
1473:
1426:
1425:
1406:
1405:
1378:
1370:
1369:
1322:
1321:
1296:
1295:
1260:
1252:
1251:
1191:
1173:
1168:
1167:
1128:
1123:
1122:
1085:Kervaire (1960)
1055:
1050:
1049:
1024:
1023:
1002:
997:
996:
978:
977:
951:
946:
945:
900:
881:
876:
875:
865:
823:
822:
786:
758:
753:
752:
711:
710:
680:
675:
674:
656:
655:
626:
621:
620:
590:
585:
584:
566:
565:
536:
531:
530:
413:
402:
401:
364:
363:
352:
335:
304:
303:
254:
249:
248:
210:
209:
181:
165:
160:
159:
137:
136:
102:
97:
96:
68:Michel Kervaire
25:
24:
12:
11:
5:
2885:
2883:
2875:
2874:
2872:Surgery theory
2869:
2859:
2858:
2855:
2854:
2848:
2831:
2818:
2815:
2814:
2813:
2810:Exotic spheres
2807:
2801:
2796:
2791:
2784:
2783:External links
2781:
2779:
2778:
2737:
2711:
2698:
2673:
2655:
2637:
2620:(3): 397–413,
2600:
2575:
2549:
2529:(3): 349–369.
2509:
2483:(3): 504–537.
2458:
2416:
2372:
2354:
2341:
2321:
2303:(1): 157–186.
2287:
2259:(3): 533–550.
2242:
2212:
2209:
2208:
2186:
2185:
2183:
2180:
2179:
2178:
2168:
2156:
2153:
2138:
2134:
2130:
2126:
2122:
2117:
2113:
2109:
2105:
2079:
2075:
2071:
2066:
2044:
2040:
2036:
2031:
2010:
2007:
2006:
2005:
2004:
2003:
1996:
1989:
1965:
1955:
1944:Browder (1969)
1941:
1935:
1921:
1915:
1907:
1904:
1880:
1876:
1871:
1855:Michael Atiyah
1839:
1836:
1833:
1813:
1810:
1807:
1790:, and
1775:
1772:
1767:
1763:
1727:
1724:
1719:
1715:
1682:
1679:
1675:isotropy index
1667:Hopf fibration
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1551:
1548:
1522:
1516:
1513:
1511:
1508:
1505:
1504:
1501:
1498:
1496:
1493:
1492:
1490:
1472:
1469:
1448:
1445:
1442:
1439:
1436:
1433:
1413:
1391:
1388:
1385:
1381:
1377:
1366:J-homomorphism
1339:
1334:
1330:
1309:
1306:
1303:
1273:
1270:
1267:
1263:
1259:
1248:
1247:
1235:
1232:
1228:
1222:
1217:
1213:
1209:
1204:
1201:
1198:
1194:
1190:
1186:
1180:
1176:
1135:
1131:
1111:exotic spheres
1068:
1065:
1062:
1058:
1037:
1034:
1031:
1009:
1005:
985:
964:
961:
958:
954:
932:
928:
924:
919:
915:
912:
907:
903:
899:
894:
891:
888:
884:
873:homotopy group
869:Lev Pontryagin
864:
861:
848:
845:
842:
839:
836:
833:
830:
808:
805:
802:
799:
796:
793:
789:
785:
780:
777:
774:
771:
768:
765:
761:
749:Hopf invariant
736:
733:
730:
727:
724:
721:
718:
696:
693:
690:
687:
683:
663:
642:
639:
636:
633:
629:
606:
603:
600:
597:
593:
573:
552:
549:
546:
543:
539:
497:
496:
485:
481:
477:
473:
468:
464:
461:
457:
453:
449:
444:
440:
437:
434:
429:
426:
423:
420:
416:
412:
409:
385:
381:
377:
372:
360:quadratic form
351:
348:
321:
316:
312:
270:
267:
264:
261:
257:
232:
229:
226:
223:
220:
217:
191:
188:
184:
180:
175:
172:
168:
147:
144:
118:
115:
112:
109:
105:
87:homology group
47:
44:
41:
38:
35:
32:
13:
10:
9:
6:
4:
3:
2:
2884:
2873:
2870:
2868:
2865:
2864:
2862:
2852:
2849:
2845:
2841:
2837:
2832:
2830:
2829:
2824:
2821:
2820:
2816:
2811:
2808:
2805:
2802:
2800:
2797:
2795:
2792:
2790:
2787:
2786:
2782:
2776:
2772:
2768:
2764:
2759:
2754:
2750:
2746:
2742:
2738:
2735:
2731:
2726:
2721:
2717:
2712:
2709:
2705:
2701:
2695:
2691:
2687:
2683:
2679:
2674:
2671:
2667:
2666:
2661:
2656:
2653:
2649:
2648:
2643:
2638:
2635:
2631:
2627:
2623:
2619:
2615:
2614:
2609:
2605:
2601:
2597:
2593:
2592:
2584:
2580:
2576:
2573:
2569:
2564:
2559:
2555:
2550:
2546:
2542:
2537:
2532:
2528:
2524:
2523:
2518:
2514:
2510:
2506:
2502:
2498:
2494:
2490:
2486:
2482:
2478:
2471:
2467:
2463:
2459:
2455:
2451:
2447:
2443:
2439:
2435:
2431:
2427:
2426:
2421:
2417:
2413:
2409:
2404:
2399:
2395:
2391:
2390:
2385:
2381:
2377:
2373:
2370:
2366:
2365:
2360:
2355:
2352:
2348:
2344:
2338:
2334:
2330:
2326:
2322:
2318:
2314:
2310:
2306:
2302:
2298:
2297:
2292:
2288:
2284:
2280:
2276:
2272:
2267:
2262:
2258:
2254:
2253:
2248:
2243:
2239:
2235:
2231:
2227:
2223:
2219:
2214:
2213:
2206:
2205:
2200:
2196:
2191:
2188:
2181:
2176:
2172:
2169:
2166:
2162:
2159:
2158:
2154:
2152:
2136:
2132:
2128:
2124:
2120:
2115:
2111:
2107:
2103:
2094:
2073:
2069:
2038:
2034:
2020:
2016:
2008:
2001:
1997:
1994:
1990:
1988:
1984:
1983:
1981:
1977:
1973:
1969:
1966:
1963:
1959:
1956:
1953:
1949:
1945:
1942:
1939:
1936:
1933:
1929:
1925:
1922:
1919:
1916:
1913:
1910:
1909:
1905:
1903:
1900:
1896:
1878:
1874:
1861:
1856:
1851:
1837:
1834:
1831:
1811:
1808:
1805:
1797:
1793:
1789:
1773:
1770:
1765:
1761:
1752:
1748:
1744:
1739:
1725:
1722:
1717:
1713:
1704:
1700:
1696:
1692:
1688:
1680:
1678:
1676:
1672:
1671:Arf invariant
1668:
1652:
1649:
1643:
1640:
1637:
1631:
1611:
1608:
1602:
1599:
1596:
1590:
1587:
1581:
1578:
1575:
1569:
1549:
1546:
1538:
1520:
1514:
1509:
1506:
1499:
1494:
1488:
1478:
1470:
1468:
1466:
1462:
1446:
1443:
1440:
1437:
1434:
1431:
1411:
1389:
1386:
1383:
1379:
1375:
1367:
1363:
1359:
1355:
1337:
1332:
1328:
1307:
1304:
1301:
1294:of dimension
1293:
1289:
1271:
1268:
1265:
1261:
1257:
1233:
1230:
1226:
1220:
1215:
1211:
1202:
1199:
1196:
1192:
1188:
1184:
1178:
1166:
1165:
1164:
1162:
1160:
1154:
1152:
1133:
1120:
1116:
1112:
1108:
1104:
1102:
1098:
1094:
1090:
1086:
1082:
1066:
1063:
1060:
1056:
1035:
1032:
1029:
1007:
1003:
962:
959:
956:
952:
926:
922:
913:
905:
901:
892:
889:
886:
882:
874:
870:
862:
860:
846:
843:
840:
837:
834:
831:
828:
806:
803:
800:
797:
794:
791:
787:
778:
775:
772:
769:
766:
763:
759:
750:
734:
731:
728:
725:
722:
719:
716:
694:
691:
688:
685:
681:
640:
637:
634:
631:
627:
604:
601:
598:
595:
591:
550:
547:
544:
541:
537:
529:
525:
521:
516:
514:
511:of the usual
510:
506:
502:
483:
475:
471:
451:
447:
438:
435:
427:
424:
421:
418:
414:
410:
407:
400:
399:
398:
379:
375:
361:
357:
356:Arf invariant
349:
347:
345:
339:
319:
314:
310:
301:
297:
293:
288:
284:
268:
265:
262:
259:
255:
246:
243:-dimensional
227:
224:
221:
218:
207:
189:
186:
182:
178:
173:
170:
166:
145:
142:
134:
116:
113:
110:
107:
103:
95:
92:
88:
84:
80:
79:Arf invariant
75:
73:
69:
65:
61:
58:-dimensional
42:
39:
36:
33:
23:
19:
2835:
2826:
2748:
2744:
2715:
2677:
2663:
2645:
2617:
2611:
2598:(6): 804–809
2595:
2589:
2553:
2526:
2520:
2480:
2476:
2429:
2423:
2396:(1): 1–262.
2393:
2387:
2362:
2328:
2300:
2294:
2256:
2250:
2221:
2217:
2204:MathOverflow
2202:
2190:
2174:
2164:
2092:
2014:
2012:
1999:
1992:
1986:
1979:
1975:
1971:
1951:
1947:
1931:
1927:
1852:
1740:
1702:
1698:
1694:
1690:
1686:
1684:
1474:
1464:
1361:
1353:
1287:
1249:
1158:
1150:
1118:
1114:
1105:
1088:
1083:
866:
519:
517:
500:
498:
353:
291:
289:
285:
244:
90:
76:
17:
15:
2432:: 257–270.
1097:PL manifold
204:), and the
2861:Categories
2741:Xu, Zhouli
2182:References
1689:there are
1153:-cobordism
528:immersions
350:Definition
247:invariant
64:surgically
2758:1410.6199
2725:1001.4751
2670:EMS Press
2652:EMS Press
2563:1104.4523
2454:120977898
2403:0908.3724
2369:EMS Press
2261:CiteSeerX
2161:Signature
2121:×
1962:Xu (2016)
1835:≥
1809:≥
1771:−
1723:−
1507:−
1329:π
1212:π
1208:→
1175:Θ
1157:homotopy
1130:Θ
1033:≥
984:→
883:π
784:→
720:≠
662:→
572:→
463:→
411::
300:Zhouli Xu
245:symmetric
179:≅
133:signature
91:quadratic
72:Cahit Arf
2581:(2011),
2522:Topology
2468:(1963).
2327:(1972),
2155:See also
1471:Examples
1161:-spheres
1099:with no
944:of maps
751:of maps
60:manifold
2775:3523064
2730:Bibcode
2708:2498881
2634:1970764
2616:, (2),
2568:Bibcode
2545:0214072
2505:0148075
2497:1970128
2446:0139172
2351:0358813
2317:1970686
2283:0810962
2238:1970470
2195:comment
1930:+2 for
1906:History
1794: (
1745: (
1352:is the
863:History
507:) is a
358:of the
94:L-group
81:of the
2836:Nature
2773:
2706:
2696:
2632:
2543:
2503:
2495:
2452:
2444:
2349:
2339:
2315:
2281:
2263:
2236:
2173:, a (4
2091:. For
1360:, and
1250:where
22:framed
2753:arXiv
2720:arXiv
2630:JSTOR
2586:(PDF)
2558:arXiv
2493:JSTOR
2473:(PDF)
2450:S2CID
2398:arXiv
2313:JSTOR
2255:. 2.
2234:JSTOR
2163:, a 4
1934:>1
1477:torus
1022:(for
821:(for
709:(for
2694:ISBN
2337:ISBN
2013:The
1796:2016
1747:1969
1404:and
290:The
208:, a
135:, a
2840:doi
2763:doi
2686:doi
2622:doi
2531:doi
2485:doi
2434:doi
2408:doi
2394:184
2305:doi
2271:doi
2226:doi
2201:",
2021:to
1838:254
1356:th
1148:of
859:).
342:. (
2863::
2838:.
2771:MR
2769:,
2761:,
2749:20
2747:,
2728:,
2718:,
2704:MR
2702:,
2692:,
2680:,
2668:,
2662:,
2650:,
2644:,
2628:,
2618:94
2606:;
2596:58
2594:,
2588:,
2566:,
2541:MR
2539:.
2525:.
2519:.
2501:MR
2499:.
2491:.
2481:77
2479:.
2475:.
2464:;
2448:.
2442:MR
2440:.
2430:34
2428:.
2406:.
2392:.
2382:;
2378:;
2367:,
2361:,
2347:MR
2345:,
2331:,
2311:.
2301:90
2299:.
2279:MR
2277:.
2269:.
2257:30
2232:.
2222:83
1320:,
346:)
283:.
74:.
2846:.
2842::
2765::
2755::
2732::
2722::
2688::
2624::
2570::
2560::
2547:.
2533::
2527:6
2507:.
2487::
2456:.
2436::
2414:.
2410::
2400::
2319:.
2307::
2285:.
2273::
2240:.
2228::
2175:k
2165:k
2137:2
2133:/
2129:n
2125:S
2116:2
2112:/
2108:n
2104:S
2093:n
2078:Z
2074:2
2070:/
2065:Z
2043:Z
2039:2
2035:/
2030:Z
2000:n
1993:n
1987:n
1980:k
1976:n
1972:n
1964:.
1954:.
1952:2
1948:n
1932:n
1928:n
1879:2
1875:P
1870:O
1832:n
1824:(
1812:8
1806:k
1774:2
1766:k
1762:2
1726:2
1718:k
1714:2
1703:n
1699:n
1691:n
1687:n
1653:1
1650:=
1647:)
1644:1
1641:,
1638:1
1635:(
1632:Q
1612:0
1609:=
1606:)
1603:1
1600:,
1597:0
1594:(
1591:Q
1588:=
1585:)
1582:0
1579:,
1576:1
1573:(
1570:Q
1550:y
1547:x
1521:)
1515:0
1510:1
1500:1
1495:0
1489:(
1465:n
1447:3
1444:+
1441:k
1438:4
1435:=
1432:n
1412:J
1390:1
1387:+
1384:n
1380:P
1376:b
1362:J
1354:n
1338:S
1333:n
1308:1
1305:+
1302:n
1288:n
1272:1
1269:+
1266:n
1262:P
1258:b
1234:,
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